PHYSICAL REVIEW B 1 FEBRUARY 1999-IIVOLUME 59, NUMBER 6

Anisotropic magnetoresistance in the organic superconductorb9-„BEDT-TTF …2SF5CH2CF2SO3

X. Su and F. ZuoDepartment of Physics, University of Miami, Coral Gables, Florida 33124

J. A. Schlueter and Jack M. WilliamsChemistry and Materials Science Divisions, Argonne National Laboratory, Argonne, Illinois 60439

P. G. Nixon, R. W. Winter, and G. L. GardDepartment of Chemistry, Portland State University, Portland, Oregon 97207

~Received 26 June 1998; revised manuscript received 3 September 1998!

In this paper, we report transport measurements of interlayer magnetoresistance with field parallel andperpendicular to the current direction in an all organic superconductorb9-~BEDT-TTF!2SF5CH2CF2SO3. ForHi I , the isothermal magnetoresistanceR(H) at low temperatures (T<Tc) displays a peak effect as a functionof field. ForH'I , R(H) increases monotonically with increasing field. The results are very analogous to theinterlayer magnetoresistance ink-(BEDT-TTF)2X compounds. The observation of the peak effect or negativemagnetoresistance in different systems forHi I' plane suggests that it is intrinsic to the layered organicsuperconductors. ForH'I , the large positive magnetoresistance is in a general agreement with a two bandmodel for charge transport.@S0163-1829~99!07405-6#

ces-

str

ds,

i-nalv

erteane

ara

rgn

or

sIF

0llicns1

-

ly

yerlarn-

t-

to

s

Forith a

bedre-

s-

er

-is

st ofon-gold

sut

I. INTRODUCTION

Interlayer transport in layered systems has been of reinterest.1–7 In anisotropic cuprates such aBi2Sr2CaCu2O81x , interlayer resistivity exhibits a semiconducting temperature dependence, while the in-plane resiity is metallic. With applied field perpendicular to the supeconducting layers, the semiconducting behavior is pushea lower temperature.4,6 In layered organic superconductorespecially the k-~BEDT-TTF!2Cu@N~CN!2#Br andk-~BEDT-TTF!2Cu~NCS!2 salts, interlayer resistance is typcally three orders of magnitude larger than the in-plaresistivity.8,9 However, the temperature dependence is qutatively similar, i.e., semiconducting for temperature aboabout 100 K and metallic for temperature below it. Furthmore, interlayer transport in these materials has shown inesting field and temperature-dependent magnetoresistpeak effect.10–14 Various models including vortex-latticinteraction,10 presence of magnetic impurities,11 stackedJosephson-junction model with a field dependent quasipcle tunneling12–14 have been proposed to explain the peeffect, however, the origin remains controversial.15

To understand the mechanism in the interlayer chatransport, we have performed transport measurement ohighly two-dimensional all organic superconductb9-~BEDT-TTF!2SF5CH2CF2SO3.

16,19,20 The b9 structurecontains layers of nearly parallel BEDT-TTF moleculewhich are canted with respect to the stacking axis.contrast thek-type structures contain orthogonal BEDT-TTmolecules. Unlike thek-(BEDT-TTF)2X, where resistivityat ambient pressure shows a broad peak at near 10both the in-plane and interlayer resistivity are metafrom room temperature down. The superconducting tration temperature is about 5 K, considerably smaller thanand 10 K for the k-~BEDT-TTF!2Cu@N~CN!2#Br andk-~BEDT-TTF!2Cu~NCS!2 salts, respectively. A compara

PRB 590163-1829/99/59~6!/4376~6!/$15.00

nt

iv--to

ei-e-r-ce

ti-k

ea

,n

K,

i-1

tive study with thek-phase superconductors is thus highdesirable.

In this paper, we report measurements of interlamagnetoresistance with field parallel and perpendicuto the current direction in the organic supercoductor b9-~BEDT-TTF!2SF5CH2CF2SO3. For Hi I , the iso-thermal magnetoresistanceR(H) displays a peak effecas a function of field. ForH'I , R(H) increases monotonically with increasing field. The results are very analogousthe interlayer magnetoresistance ink-(BEDT-TTF)2Xcompounds.10–14The similarity for the two different systemsuggests that the peak magnetoresistance forHi I is intrinsicto the layered structure of the organic superconductors.H'I , the magnetoresistance is in a general agreement wtwo band model for charge transport.

II. EXPERIMENT

Single crystals ofb9-~BEDT-TTF!2SF5CH2CF2SO3 weresynthesized by the electrocrystallization technique descrielsewhere.16 Several crystals were used in these measuments with average dimensions of 130.7830.33 mm. Ex-tensive measurements were made on one crystal withTc;5 K. The Tc is defined as the midpoint in the resitive transition. Depending on the cooling rate,Tc can bevaried from;5 to 5.5 K. The room-temperature interlayresistivity is about 700V cm and the in-plane resistivityabout 0.2V cm. The resistivity ratio between room temperature and superconducting transition temperature@r'~300 K!/r'~6 K!#;230. The interlayer resistance wameasured with use of the four-probe technique. Contacthe gold wires to the sample was made with a Dupont cducting paste. Typical contact resistances between thewire and the sample were about 10V. A current of 1mA wasused to ensure linearI -V characteristics. The voltage wadetected with a lock-in amplifier at low frequencies of abo

4376 ©1999 The American Physical Society

sulle

waib

v-

edldld

thivvit

eain.t

eaat

peorgaes

at

era-

em-

tionn-he

reld

d

PRB 59 4377ANISOTROPIC MAGNETORESISTANCE IN THE . . .

312 Hz. The samples were cooled slowly to below theperconducting transition temperature with the field paraand perpendicular to the crystallographicb axis. To avoidpressure effect due to solidification of grease, the samplemechanically held by thin gold wires. Because of the smsample size, a misalignment of up to 5 degrees was poss

III. RESULTS AND DISCUSSIONS

Shown in Fig. 1 is a typical plot of the interlayer resistiity as a function of field atT51.82 K. The midpoint ofresistive transition is about 5.5 K for the sample studiClearly, the resistivityr starts to rise rapidly at an onset fieof about 0.2 T.r reaches a peak value at a peak fieHpeak;0.7 T. For field greater thanHpeak, the resistivitydecreases sharply with increasing field. At about 3 T,slopedR/dH changes sign and becomes small and positas shown in the inset. If we compare the values of resistiat 3 T and at the peak, it is easy to see@rpeak/r(3 T)#.2.

Figure 2 shows the overlay of the resistivity peak asfunction of field at various temperatures,T51.82, 2, 2.2, 2.6,3, 3.5, 4, 4.5, and 5 K. With increasing temperature, the pshifts toward zero field and the magnitude of the peakcreases initially and reaches a maximum at around 2.6 Kdecreases with further increase in temperature. Plotted ininset is the peak field versus the temperature.Hpeak increasesmonotonically with decreasingT. The solid line is a fit toHpeak5Ho(12T/Tc)

n, with Ho51.560.2 T, n52.160.1,Tc55.460.2 K. The temperature dependence of the pfield demonstrates clearly that the peak effect is associwith the superconductivity, since it disappears aboveTc .The temperature dependence ofHpeak(T) in the temperaturerange investigated is reminiscent of the temperature dedence of the critical fields in the cuprate superconductThe characteristic field where the magnetoresistance is apositive decreases with increasing temperature. For tempture aboveTc , magnetoresistance is always small and potive.

FIG. 1. Interlayer resistivity as a function of field forH'planeat T51.82 K. The inset is an expanded view ofr(H) at high fields.

-l

asllle.

.

ee,y

a

k-Ithe

ked

n-s.in

ra-i-

Plotted in Fig. 3 is an overlay of interlayer resistivityH50 and 2 T, in comparison withrpeak(T). At H52 T, theresistivity decreases monotonically with decreasing tempture. The offset forr(H52 T) aboveTc from the zero-fielddata is due to the positive magnetoresistance. At lower tperatures, therpeak(T) is clearly displaced from ther(H52 T) curve, and reaches a maximum at around 2.6 K.

Figure 4 is an overlay of magnetoresistance as a funcof field at various temperatures for field parallel to the coducting plane. Unlike for field perpendicular to the plane, tmagnetoresistance displays several different features:~1! the

FIG. 2. Interlayer resistivity peak as a function of field foH'plane at various temperatures. The inset is a plot of peak fiversus temperature and the line is a fit.

FIG. 3. Overlay of the peak resistivity, resistivity at 2 T, anzero-field resistivity versus temperature.

eis

atw

eo

in

gceca

tod

T

teetw0

re-

hedenceof

-the

k isen-

f

er-

4378 PRB 59X. SU et al.

resistive onset field is much larger forHiplane, for example,Honset;5 T at 2 K, compared toHonset;0.15 T forH'plane; ~2! the rise of resistivity forH.Honset is muchslower in theHiplane case;~3! there is no peak in resistivityas a function of field. Instead,r(H) increases monotonicallywith increasing field;~4! the magnetoresistance at largfields for Hiplane is considerably larger than the peak restivity value for H'plane, similar to thek-phase salts.17,18 Itshould be noted that since the measurements for field parand perpendicular to the conducting plane were done inseparate runs, a larger cooling rate for theH'plane has re-sulted in a larger normal-state resistivity and slightly reducTc . This is analogous to the cooling rate dependenceserved in thek-~BEDT-TTF!2Cu@N~CN!2#Br.21 If we nor-malize the zero-field resistivity aboveTc to that ofHiplaneby a scaling factor@ri~10 K!/r'~10 K!#>0.68, the normal-ized peak resistivity at 3 K is r'

norm;6.530.6854.4V cmcompared withr i~8 T!;9V cm.

At T.Tc , magnetoresistance increases with increasfield. Shown in Fig. 5 is an overlay ofr(H) at various tem-peraturesT57.75, 9.5, 12, 14, 16, and 18 K. With increasintemperature, the field dependence of magnetoresistanincreasingly smaller, as is clear from the figure. The databe well fitted tor(H)5ro1Dr (2)H21Dr (4)H4, with Dr (2)

and Dr (4) being the coefficients forH2 and H4 terms, re-spectively.

The temperature and field dependence of the magnesistance can be summarized by plotting the temperaturependence of the fittedDr (2) andDr (4) values scaled toro ,as shown in Fig. 6 in a semilog scale. Clearly, (Dr (2)/ro)increases exponentially with decreasing temperature.solid line is a fit to (Dr (2)/ro)50.163102T/8. Dr (4) isalways negative for the temperature range investiga2(Dr (4)/ro) is plotted against temperature in the insAgain, an exponential temperature dependence is seen(Dr (4)/ro)520.063102T/5.4. For temperature above 2

FIG. 4. Interlayer resistivity as a function of field forHiplane atlow temperatures. The temperature increment is 0.5 K.

-

llelo

db-

g

isn

re-e-

he

d,.ith

K, Dr (4) term has a negligible contribution to the magnetosistance.

An alternative way to look at the field dependence of tmagnetoresistance is to measure the temperature depenof resistivity at a fixed field. Shown in Fig. 7 is an overlaythe r(T) at various applied fieldsH50, 1, 2, 4, 6, and 8 Tfor Hiplane. ForT.6 K, r(T,H) increases monotonicallywith T. For T,Tc , r(T,H) has a maximum at a field dependent peak temperature. The results are equivalent toisothermal field dependence in Fig. 4, except here the peabetter defined due to small temperature increments. As mtioned earlier, the zero-field resistivity forHiplane is smallerthan that forH'plane and theTc is about 0.2 K higher. Thecooling rate dependence of the resistivity and the reducedTcwith increasing r are analogous to that o

FIG. 5. Interlayer resistivity as a function of field forHiplane athigh temperatures.

FIG. 6. The extrapolated field and temperature coefficients vsus temperature in a semilog scale.

y

-rg

nthis

tival

es

pem

n

ionllegtiomreret

roan

sirte

iv

or-illtateredis-

entlya

y a

-

-y

, the

-Thek in

id-

is

-

ume

es

-e

PRB 59 4379ANISOTROPIC MAGNETORESISTANCE IN THE . . .

k-~BEDT-TTF!2Cu@N~CN!2#Br salt.21

The field and temperature dependence of the interlaresistivity in the b9-~BEDT-TTF!2SF5CH2CF2SO3 is verysimilar to that of k-(BEDT-TTF)2X with X5Cu@N~CN!2#Br and Cu~NCS!2, even though the superconducting transition temperature in thek phase is much largeat 11 and 10 K, respectively. The peak effect or large netive magnetoresistance in theHi I'plane is still controver-sial. One possibility is that the negative magnetoresistaarises from the presence of magnetic impurities insamples.11 With increasing field, the magnetic scatteringsuppressed, thus leading to a decrease in magneto-resisHowever, the presence of magnetic impurities in theorganic b9-~BEDT-TTF!2SF5CH2CF2SO3 is very unlikely,because it has no metal atoms, such as the Cu(II ) in thek-phase structure. The absence of negative magnetortance of the in-plane resistivity for high qualityk-phasesamples, where peak effect in the interlayer resistancesists, suggests strongly that the magnetoresistance peakbe intrinsic to the layered systems.13 This is also supportedby a recent study of magnetoresistance peak as a functioinhomogeneities in the k-~BEDT-TTF!2Cu@N~CN!2#Brsalts.14 For samples with large superconducting transitwidths, the peak effect disappears. With increasingly smatransition width the peak becomes more pronounced. Netive magnetoresistance can also arise from weak localizaand electron-electron interactions as observed in manytallic systems.22 Disorders can lead to negative magnetosistance because the magnetic field disrupts the cohebackscattering and suppresses the localization. Similarly,magnetic field will decrease the attractive electron-electinteraction and lead to a smaller resistance for charge trport. However, the magnitude of the peak or (Dr/r);1 istoo large to be considered for this model. A magnetoretance peak is also possible if one assumes a large volattice interaction.10 In this model, vortex-lattice interactionleads to local disorders in the electronic potentials, thus g

FIG. 7. Overlay of interlayer resistivity as a function of temperature for various applied field forHiplane.

er

a-

cee

ity.l-

is-

r-ay

of

ra-n

e--nt

hens-

s-x-

-

ing rise to an extra scattering. With increasing field, the vtex cores start to overlap. Eventually, the large field wreduce the lattice distortions and resume the normal-selectric conductivity. Although the model is plausible, thehas been no report of structural evidence for the latticetortions.

Negative magnetoresistance has been discussed recin terms of stacked Josephson-junction model includingfield dependent quasiparticle tunneling.12–14 In this model,the resistive transition at small fields can be described bresistively shunted Josephson junction withR(H)5Rn@ I o(\I c/2ekT)#22, whereRn is the normal-state resistance,\ is the Planck’s constant,I c is the critical current,e isthe charge of an electron, andI o is the modified Bessel function. ForH'plane, the junction is effectively determined bthe distance between the neighboring vortices. In generalquasiparticle conductanceYss is thermally activatedYss;exp@2D(T,H)/kT#, and the pair conductance isYp;@ I o(\I c/2ekT)#221 The total conductance isY5Yss1Yp51/r. With increasing field, the pair contribution decreases while the quasiparticle contribution increases.competition among the two terms naturally leads to a peathe conductance or magnetoresistance.

To analyze it quantitatively, charge transport is consered to be along an effective Josephson junction of areaa2

'(FoH1Ho) between the densely packed vortices,4 Ho be-ing a fitting parameter. The total junction conductancegiven by Y5Y1„$I o@eJFo /(H1Ho)2kT#%221…1Y2(H)exp@2D(T,H)/kT#, here eJ5(\I c/2ea2) defines anintrinsic Josephson coupling energy,Y151/Rn . If the fielddependence of theeJ and D(T,H) is considered to be proportional to (12@H/Hc2(T)#2) and assume a constantY2 ,we are unable to get a reasonable fit to the data. If we ass1/Y2(H)5r0@11(Ho/H)n#, a nearly perfect fit near thepeak can be obtained, as shown in Fig. 8. The fit giv1/Y156.960.1V cm, (eoJFo/2kT)50.4660.02T, Ho520.0360.005T, 1/Y2(H)52.341(1.53/H2) V cm,(Do /kT)5060.01, Hc250.9T for r~H! at T52.2 K.

FIG. 8. Interlayer resistivity as a function of field at 2.2 K. Thline is a fit to the data.

uhetetiv

reesO

tein

eThn

tivyigpo

ic

ieae

ear

re

btoThu

ieon

e

-

m-

duei-

ns-tor

func-m-

r in-nce.ilar

on-nickedasi-sis-ergy

ar-ved

om-

ne.

-ryof

un-ate15.

4380 PRB 59X. SU et al.

Clearly, the exponential term in the quasiparticle contribtion is negligible. Fits without the exponential term give tsame parameters. Similar results are obtained for otherperatures. This is consistent with the fact that the negamagnetoresistance extends well aboveHc2 . Unless a verydifferent field dependence of the gap energy is considesuch asD(T,H);1/Ha, a simple model considered abovfails to support the picture where the negative magnetoretance comes from the enhanced quasiparticle tunneling.possibility is to include the fluctuation effect forH.Hc2 inthe treatment of quasiparticle contribution. It should be nothat in a similar approach to fit the peak effectk-~BEDT-TTF!2Cu~NCS!2, the model failed to fit the high-field data.12

For field applied parallel to the plane, the field will bmostly confined in between the superconducting layers.critical field in this direction is much larger than that whethe field is perpendicular to the plane. The larger resisonset field is a direct consequence of the large anisotropthis material. The large positive magnetoresistance at hfield and high temperatures may be associated with the osheets and closed pockets in the Fermi surface. For mmetals, the magnetoresistance is negligible with a typ(Dr/r) in the order of (vt)25@(eH/m)t#2. For example,in coppervt;531023 for a field of 1 T. A similar estimatefor the title compound would yieldvt5A(Dr (2)/ro);0.2at 1 T. This is almost two orders of magnitude larger thanconventional metals. However, the interlayer resistivity nthe transition is about 1V cm, about six orders of magnitudlarger than in copper. The large (Dr/r) demonstrates thegross inadequacy of a single band picture. In the presenctwo bands, as in the case of most organic conductors, a lpositive magnetoresistance is foreseeable.23,24 Considern1andn2 as the charge densities for the two bands andm1 andm2 as the carrier mobilities, respectively. The zero-fieldsistivity is ro5(n1m11n2m2)21. At low fields, the trans-verse magnetoresistance is (Dr/ro)5@n1n2m1m2(m12m2)2H2/(n1m11n2m2)2#; at higher fields a negativeH4

term should be included in the two band model. It shouldnoted that the above expression for the transverse magnesistance applies only to the isotropic two band system.anisotropic nature of the present compound, added withcertainties in the charge carrier densities and mobilitmakes it very complicated task to identify the contributifrom each terms. Nevertheless, if we assumem1@m2 , then

.

,

,

-

m-e

d,

is-ne

d

e

einh

enst

al

nr

ofge

-

ere-en-s,

(Dr/ro)'(n2m1m2 /n1)H2. The exponential temperaturdependence in the (Dr (2)/ro)50.163102T/8 for H51 Tsuggests thatm1m2;102T/8. A similar exponential temperature dependence might be expected in (Dr (4)/ro);102T/5.4, since it involves higher orders ofm1m2 . While aconventional Fermi-liquid system gives a power-law teperature dependence in (Dr (2)/ro), the experimental resultsmay suggest the exponential temperature dependence iseither to complications of in-plane anisotropy or non-Fermliquid behavior.

IV. CONCLUSIONS

In summary, we have reported a detailed interlayer traport measurement in the all-organic superconducb9-~BEDT-TTF!2SF5CH2CF2SO3. For field perpendicular tothe layers, a peak in magnetoresistance is observed as ation of field. The peak field increases with decreasing teperature withHpeak(T)5Ho(12T/Tc)

2. For H.Hpeak, alarge negative magnetoresistance is observed. Furthecrease in field results in a small positive magnetoresistaThe interlayer magnetoresistance peak effect is very simto that of thek-phase organic superconductors. This demstrates that the peak effect is intrinsic to the layered orgasuperconductors. Quantitative analysis in terms of stacJosephson-junction model plus a thermally activated quparticle tunneling is unable to fit the negative magnetoretance, assuming a simple field dependence of the gap enD(T,H)5Do$12@H/Hc2(T)#2%. The origin of the peak ef-fect in the interlayer transport remains unclear. For field pallel to layers, large positive magnetoresistance is obserfor all temperatures with large resistive onset field forT,Tc . Although the data are generally consistent with twband picture for the organic conductors, the exponential teperature dependence of (Dr (2)/ro) calls for more systematicstudies of the anisotropic transport in the conducting pla

ACKNOWLEDGMENTS

The work is supported in part by NSF Grant No. DMR9623306. Work performed at Argonne National Laboratowas supported by the U.S. Department of Energy, OfficeBasic Energy Sciences, Division of Materials Sciences,der Contract No. W-31-109-ENG-38. Work at Portland StUniversity is supported by NSF Grant No. CHE-96328and the Petroleum Research Fund ACS-PRF 31099-AC1

Y.

,

s.

.

1S. Chakravarty and P. W. Anderson, Phys. Rev. Lett.72, 3859~1994!.

2A. G. Rojo and K. Levin, Phys. Rev. B48, 16 861~1993!.3N. Kumar and A. M. Jayannavar, Phys. Rev. B45, 5001~1992!.4G. Briceno, M. F. Crommie, and A. Zettl, Phys. Rev. Lett.66,

2164 ~1991!.5L. B. Ioffe, A. I. Larkin, A. A. Varlamov, and L. Yu, Phys. Rev

B 47, 8936~1993!.6A. S. Nygmatulin, A. A. Varlamov, D. V. Livanov, G. Balestrino

and E. Milani, Phys. Rev. B53, 3557~1996!.7A. A. Varlamov, G. Balestrino, E. Milani, and D. V. Livanov

Adv. Phys.~to be published!.

8G. Saito, H. Yamochi, T. Nakamura, T. Komatsu, T. Ishiguro,Nogami, Y. Ito, H. Mori, and K. Oshima, Synth. Met.41-43,1993 ~1991!.

9L. I. Buravov, N. D. Kushch, V. A. Merzhanov, M. V. OsherovA. G. Khomenko, and E. B. Yagubskii, J. Phys. I2, 1257~1992!.

10F. Zuo, J. A. Schlueter, M. E. Kelly, and J. M. Williams, PhyRev. B54, 11 973~1996!; Physica C282, 1901~1997!.

11C. H. Mielke, N. Harrison, D. G. Rickel, A. H. Lacerda, R. MVestal, and L. K. Montgomery, Phys. Rev. B56, R4309~1997!.

12S. Friemel, C. Pasquier, and D. Jerome, Physica C292, 273~1997!.

ica

M

-D.

.P

.o,

.

. B

il-H.

r, J.s.

s,

M.

ys.

PRB 59 4381ANISOTROPIC MAGNETORESISTANCE IN THE . . .

13H. Ito, T. Ishiguro, T. Komatsu, G. Saito, and H. Anzai, PhysB 201, 470 ~1994!; J. Supercond.7, 667 ~1994!.

14F. Zuo, X. Su, P. Zhang, J. A. Schlueter, M. E. Kelly, and J.Williams, Phys. Rev. B57, R5610~1998!.

15R. H. McKenzie and P. Moses~unpublished!.16U. Geiser, J. A. Schlueter, H. H. Wang, A. M. Kini, J. M. Will

iams, P. P. Sche, H. I. Zakowicz, M. L. VanZile, and J.Dudek, J. Am. Chem. Soc.118, 9996~1996!; J. A. Schlueter, U.Geiser, J. M. Williams, J. D. Dudek, M. E. Kelly, J. P. Flynn, RR. Wilson, H. I. Zakowicz, P. P. Sche, D. Naumann, T. Roy,G. Nixon, R. W. Winter, and G. L. Gard, Synth. Met.85, 1453~1997!.

17K. Murata, Y. Honda, H. Anzai, M. Tokumoto, K. Takahashi, NKinoshita, T. Ishiguro, N. Toyota, T. Sasaki, and Y. MutSynth. Met.27, A341 ~1988!.

18W. K. Kwok, U. Welp, K. D. Carlson, G. W. Crabtree, K. GVandervoort, H. H. Wang, A. M. Kini, J. M. Williams, D. L.

.

.

Stupka, L. K. Montgomery, and J. E. Thompson, Phys. Rev42, 8686~1990!.

19D. Beckmann, S. Wanka, J. Wosnitza, J. A. Schlueter, J. M. Wliams, P. G. Nixon, R. W. Winter, G. L. Gard, J. Ren, and M.-Whangbo, Eur. Phys. J. B1, 295 ~1998!.

20S. Wanka, J. Hagel, D. Beckmann, J. Wosnitza, J. A. SchlueteM. Williams, P. G. Nixon, R. W. Winter, and G. L. Gard, PhyRev. B57, 3084~1998!.

21X. Su, F. Zuo, J. A. Schlueter, M. E. Kelly, and J. M. WilliamPhys. Rev. B57, R14 056~1998!.

22P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys.57, 287~1987!.

23F. L. Pratt, A. J. Fisher, W. Hayes, J. Singleton, S. J. R.Spermon, M. Kurmoo, and P. Day, Phys. Rev. Lett.61, 2721~1988!.

24R. H. McKenzie, J. S. Qualls, S. Y. Han, and J. S. Brooks, PhRev. B57, 11 854~1998!.